8 LEBESGUE'S CONVERGENCE THEOREMS 129 from which it follows ((3.16.5) and (8.7.8)) that, for almost all x, the function f(x, z) is the limit of the sequence I/T1 Y,fkn(x)}9 where \ fc-i / _/(*, a + re2kni/n)re2kni/n Jkn\XJ "" !------2kni/n-----------"" * a + re2kni/n — z Applying (13.8.4), it follows that h(z) is the limit of the sequence and therefore that which shows that h is analytic in E (9.9.4). By replacing the term !/(( — z) in the right-hand side of (13.8.6.2) by fc!/(f-z)*+1, we obtain the formula stated for D*/z(z). (13.8.7) (i) If (An) is a sequence of integrable sets, then A = (°) An w n integrable. If also the sequence (Aw) w decreasing, we have (13.8.7.1) MA) = mfXAJ. n (ii) If(An) is a sequence of integrable sets contained in an integrable set B, then A = (J AM w integrable. n (iii) If(An) is a sequence of integrable sets such that 11=1 /on A = (J Art is integrable, and we have (13.8.7.2) »=1 If moreover the sets An are pairwise disjoint, then (13.8.7.3) /i(A)= (" complete additivity of measure *'). the right-hand side of (13.8.6.1). But it follows from the hypotheses